Question

write a two - column proof. given: m∠gfi = 131°. prove: m∠efi = 36°. statements: 1. m∠gfi = 131°. 2. m∠gfe + m∠efi = m∠gfi. 3. m∠gfe + m∠efi = 131°. 4. m∠gfe=(8x - 1)°, m∠efi = 3x°. 5. 8x - 1+3x = 131. 6. 11x - 1 = 131. 7. x = . reasons: 1. given. 2. angle addition postulate. 3. substitution property, (steps 1, 2). 4. given. 5. substitution property. 6. combine like terms. 7. addition property of equality.

Explanation:

Step1: Apply addition property of equality

Add 1 to both sides of the equation \(11x - 1=131\). So, \(11x-1 + 1=131 + 1\), which simplifies to \(11x=132\).

Step2: Solve for x

Divide both sides of the equation \(11x = 132\) by 11. So, \(\frac{11x}{11}=\frac{132}{11}\), and \(x = 12\).

Step3: Find measure of \(\angle EFI\)

Since \(m\angle EFI=3x^{\circ}\), substitute \(x = 12\) into the expression. Then \(m\angle EFI=3\times12^{\circ}=36^{\circ}\).

Answer:

1. \(11x = 132\) (after adding 1 to both sides of \(11x-1 = 131\)); \(x = 12\) (after dividing both sides of \(11x=132\) by 11) and we have proven \(m\angle EFI = 36^{\circ}\) as \(m\angle EFI=3x^{\circ}\) and when \(x = 12\), \(m\angle EFI=3\times12^{\circ}=36^{\circ}\)